Hi,
the eeprom is a small 256 byte memory chip in the xbox. So, unless you desolder it from the motherboard, you can't loose it. The chip contains information that is unique for every xbox. Part of that information is the hddkey which is used to calculate the hdd unlocking password. Due to that, the harddisk from xbox A can not be used in xbox B.
What you lost is a backup of the contents of that eeprom chip. It is a small file that contains the same information as the chip. Packages like xboxhdm use it to calculate the xbox unlocking password.
The modchip biosses are modified so that they allow the xbox to boot normally, even when there is an unlocked harddisk in the xbox. Because of that, you don't need the eeprom.bin file to make things work again.
The eeprom reader reads out the small memory chip directly. Even when the motherboard is not functionning, you can still read out the eeprom chip in most cases.
regards.